The harmonic measure of diffusion-limited aggregates including rare events

We obtain the harmonic measure of diffusion-limited aggregate (DLA) clusters using a biased random-walk sampling technique which allows us to measure probabilities of random walkers hitting sections of clusters with unprecedented accuracy; our results include probabilities as small as 10−80. We find the multifractal D(q) spectrum including regions of small and negative q. Our algorithm allows us to obtain the harmonic measure for clusters more than an order of magnitude larger than those achieved using the method of iterative conformal maps, which is the previous best method. We find a phase transition in the singularity spectrum f(α) at α≈ 14 and also find a minimum q of D(q), qmin = 0.9± 0.05. Copyright c © EPLA, 2009 Introduction. – Diffusion-limited aggregation (DLA) is a stochastic model for irreversible growth which gives rise to fractal clusters [1,2], see figs. 1, 2. The growth process is defined by releasing a random walker far from the cluster and allowing it to diffuse until it sticks to the surface and becomes part of the cluster. Then another particle is released, and so forth. The probability of sticking at various points on the cluster, i.e. the distribution of the growth probability, is a function with very large variations. It is the subject of this paper. Since the Laplace equation is equivalent to the steadystate diffusion equation, this probability distribution is proportional to the perpendicular electric field on the surface of a charged electrode with the shape of the cluster; in this context the probability is called the harmonic measure, and is defined for any surface. For fractal surfaces, including that of DLA, the harmonic measure is usually multifractal [3]. For DLA the harmonic measure is of particular interest because of the connection with the growth probability. For other fractal surfaces this connection is lost. However, the measure is still of substantial practical interest because of its relationship with physical processes such as catalysis [4]. (a)E-mail: [email protected] For many interesting equilibrium fractals the harmonic measure can be calculated using conformal field theory [5–7] or Schramm-Loewner evolution (SLE) [8]. There is no corresponding theory for DLA for which the measure must be found numerically. There are numerous studies in the literature of this quantity, for example [9–12]. This is a difficult problem because of the very large variation of the growth probability. As we will see, the dynamic range of the function is of the order of 10 even for rather small clusters. This is far out of the range accessible to straightforward random-walker sampling. In this paper, we use a biased random-walk sampling method. We can obtain extremely small growth probabilities and to accurately obtain the complete harmonic measure for DLA clusters of up to 10 particles. The method was previously used on percolation and Ising clusters [13]. For those (equilibrium) systems, we found good agreement with analytic predictions for the harmonic measure [14,15]. The harmonic measure is usually characterized in terms of the generalized dimensions D(q). For integer q, D(q) corresponds to the fractal dimension of the q point correlation function. We define D(q) by partitioning the external boundary of a DLA cluster into boxes of length l. The probability that a diffusing particle will hit the section